Will the 120cm Lencarta folding softbox fit the bill?
It's a 'proper' softbox that folds like an umbrella, Bowens S fit, so will fit the Lencarta Safari (as well as all other Lencarta lights) but the Safari is above your budget, even though it's probably the cheapest battery powered system.
The alternative would be to use a couple of hotshoe flashes fired into a softbox, but power output may be a problem, and the lighting won't be as evenly diffused of course.
Budget £300ish